forked from idrl/idrlnet
58 lines
1.9 KiB
Markdown
58 lines
1.9 KiB
Markdown
# Burgers' Equation
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Burgers' equation is formulated as following:
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$$
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\begin{equation}
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\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\nu \frac{\partial^{2} u}{\partial x^{2}}
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\end{equation}
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$$
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We have added the template of the equation into `idrlnet.pde_op.equations`.
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In this example, we take $\nu=-0.01/\pi$, and the problem is
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$$
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\begin{equation}
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\begin{array}{l}
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u_t+u u_{x}-(0.01 / \pi) u_{x x}=0, \quad x \in[-1,1], \quad t \in[0,1] \\
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u(0, x)=-\sin (\pi x) \\
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u(t,-1)=u(t, 1)=0
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\end{array}
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\end{equation}.
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$$
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## Time-dependent Domain
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The equation is time-dependent. In addition, we define a time symbol `t` and its range.
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```python
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t_symbol = Symbol('t')
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time_range = {t_symbol: (0, 1)}
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```
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The parameter range `time_range` will be passed to methods `geo.Geometry.sample_interior()` and `geo.Geometry.sample_boundary()`.
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The sampling methods generate samples containing the additional dims provided in `param_ranges.keys()`.
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```python
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# Interior domain
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points = geo.sample_interior(10000, bounds={x: (-1., 1.)}, param_ranges=time_range)
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# Initial value condition
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points = geo.sample_interior(100, param_ranges={t_symbol: 0.0})
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# Boundary condition
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points = geo.sample_boundary(100, param_ranges=time_range)
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```
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The result is shown as follows:
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## Use TensorBoard
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To monitor the training process, we employ [TensorBoard](https://www.tensorflow.org/tensorboard).
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The learning rate, losses on different domains, and the total loss will be recorded automatically.
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Users can call `Solver.summary_receiver()` to get the instance of `SummaryWriter`.
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As default, one starts TensorBoard at `./network_idr`:
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```bash
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tensorboard --logdir ./network_dir
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```
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Users can monitor the status of training:
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See `examples/burgers_equation`. |